Answer:
25 seconds
Step-by-step explanation:
we have
[tex]P(x)=-x^{2}+50x+500[/tex]
This is the equation of a vertical parabola open downward
The vertex is a maximum
The y-coordinate of the vertex is the maximum height of the projectile
The x-coordinate of the vertex is the time in seconds that the projectile takes to reach its maximum height.
Convert the quadratic equation in vertex form
we have
[tex]P(x)=-x^{2}+50x+500[/tex]
Factor -1
[tex]P(x)=-(x^{2}-50x)+500[/tex]
Complete the square
[tex]P(x)=-(x^{2}-50x+25^2)+500+25^2[/tex]
[tex]P(x)=-(x^{2}-50x+625)+500+625[/tex]
[tex]P(x)=-(x^{2}-50x+625)+1,125[/tex]
Rewrite as perfect squares
[tex]P(x)=-(x-25)^{2}+1,125[/tex]
The vertex is the point (25,1,125)
therefore
The maximum height of the projectile is 1,125 feet
The time in seconds that the projectile takes to reach its maximum height is 25 seconds